Integrating factor for first order linear equations uniqueness theorem

[gsingh2011]

My book stated the following theorem: If the functions P(x) and Q(x) are continuous on the open interval I containing the point x₀, then the initial value problem dy/dx + P(x)y = Q(x), y(x₀)=y₀ has a unique solution y(x) on I, given by the formula y=1/I(x)$\int$I(x)Q(x)dx where I(x) is the integrating factor.

Now the book showed how the Integrating Factor Method was developed, but it doesn't prove this theorem, particularly why a unique solution exists and why there are no other solutions of a different form (singular solutions).

Also it states, "The appropriate value of the constant C can be selected "automatically" by writing, I(x)=exp($\int$$^{x_{0}}_{x}$P(t)dt) and y(x)=1/I(x)[y₀+$\int$$^{x_{0}}_{x}$I(t)Q(t)dt]"

I don't understand how they got this form. If you have a definite integral there shouldn't be constant like y₀ either...

 

[Hootenanny]

As far as the proof of existence an uniqueness goes, it is straightforward and can be found at: http://arapaho.nsuok.edu/~okar-maa/news/okarproceedings/OKAR-2006/finan.pdf

As for the definite form of the integrating factor, you are indeed correct, no constant arises from the definite integral and therefore we must add an appropriate constant so that the solution satisfies the boundary conditions.

We have

$$y(x) = \frac{1}{I(x)}\left(y_0 + \int_x^{x_0} I(t)Q(t)\;\text{d}t\right)$$

$$I(x) = \exp\left(\int_x^{x_0} P(t)\;\text{d}t\right)$$

So, if we consider $x=x_0$ we find$$y(x_0) = \frac{1}{I(x_0)}\left(y_0 + \int_{x_0}^{x_0} I(t)Q(t)\;\text{d}t\right)$$

Obviously the integral vanishes, leaving

$$y(x_0) = \frac{y_0}{I(x_0)}$$

with

$$I(x_0) = \exp\left(\int_{x_0}^{x_0} P(t)\;\text{d}t\right) = \exp(0) = 1$$

Hence

$$y(x_0) = y_0$$

as required.

 

[gsingh2011]

Thanks, that makes sense. One thing I don't understand in the link you gave me is on page four, they state w(t)=y₁(t₀)-y₂(t₀)=y₀-y₀. Why are the intial values of those two functions the same, if we assumed that the two functions would be different?

 

[Hootenanny]

Thanks, that makes sense. One thing I don't understand in the link you gave me is on page four, they state w(t)=y₁(t₀)-y₂(t₀)=y₀-y₀. Why are the intial values of those two functions the same, if we assumed that the two functions would be different?

Because even though they may be different functions, if they are solutions of the same boundary value problem, then they must satisfy the same boundary conditions!

Does that make sense?

 

[blue_raver22]

I would say that the uniqueness theorem for first order linear equations is a fundamental result in mathematics that has been proven and accepted by the scientific community. It guarantees that for a given initial value problem, there exists a unique solution that can be expressed using the integrating factor method.

The proof of this theorem involves using fundamental concepts from calculus, such as the fundamental theorem of calculus and the existence and uniqueness theorem for ordinary differential equations. It also relies on the continuity of the functions P(x) and Q(x) on the open interval I containing the point x0.

The form of the integrating factor, I(x) = exp(∫P(t)dt), and the solution y(x) = 1/I(x)[y0+∫I(t)Q(t)dt] is derived using the integrating factor method, which is a standard technique for solving first order linear equations. This method involves multiplying both sides of the equation by the integrating factor, which simplifies the equation and makes it easier to solve.

The constant C in the solution comes from the fact that when we integrate a function, we always add a constant of integration. In the case of a definite integral, this constant can be chosen automatically by specifying the initial condition y(x0)=y0. This is because when x=x0, the definite integral becomes y(x0)=y0+∫I(t)Q(t)dt, which allows us to determine the value of the constant y0.

In conclusion, the uniqueness theorem for first order linear equations is a well-established result and the form of the integrating factor and solution are derived using standard mathematical techniques. The constant in the solution can be determined automatically by specifying the initial condition.